K theory of projectionless C*-algebras

Question

Is it possible to have a projectionless C*- algebra with non trivial K-theory? If so what would be such an example? I can't come up with any.

p.s.

By projectionless I mean non-unital aswell.

Answer 1

One example is given by $C_0(\mathbb R^2)$. This has no non-zero projections, and by Bott periodicity, we have \begin{align*} K_n(C_0(\mathbb R^2))=K_n(\mathbb C)=\left\{ \begin{array}{ll} \mathbb Z &:n=0,\\ 0 &:n=1. \end{array} \right. \end{align*}